Viscous energy dissipation in slender channels with porous or semipermeable walls

TL;DR

This paper derives simple formulas for viscous energy dissipation in long channels with porous walls, extending classical results to account for flow and pressure variations, with applications to osmotic flows.

Contribution

It provides generalized closed-form expressions for viscous dissipation in channels with porous or semipermeable walls, incorporating variable flow and pressure profiles.

Findings

Derived formulas for energy dissipation in porous-walled channels.

Extended classical dissipation expressions to variable flow and pressure.

Applied results to osmotic flow scenarios.

Abstract

We study the viscous dissipation in pipe flows in long channels with porous or semipermeable walls, taking into account both the dissipation in the bulk of the channel and in the pores. We give simple closed form expressions for the dissipation in terms of the axially varying flow rate $Q(x)$ and the pressure $p(x)$, generalizing the well known expression $\dot W=Q\,\Delta p$ for the case of impenetrable walls with constant $Q$ and a pressure difference $\Delta p$ between the ends of the pipe. When the pressure $p_0$ outside the pipe is constant, the result is the straightforward generalization $\dot W=\Delta \left[(p-p_0) \,Q\right]$. Finally, applications to osmotic flows are considered.